High School

If [tex]$f(4)=246.4$[/tex] when [tex]$r=0.04$[/tex] for the function [tex]$f(t)=P e^{rt}$[/tex], then what is the approximate value of [tex]$P$[/tex]?

A. 210
B. 50
C. 289
D. 1220

Answer :

To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex], given that [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex], follow these steps:

1. Substitute the known values into the equation:

We know:
- [tex]\( t = 4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( f(4) = 246.4 \)[/tex]

So, we have the equation:
[tex]\[
f(4) = P \cdot e^{0.04 \cdot 4}
\][/tex]
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]

2. Solve for [tex]\( P \)[/tex]:

To find [tex]\( P \)[/tex], you need to isolate [tex]\( P \)[/tex] on one side of the equation. This can be done by dividing both sides by [tex]\( e^{0.16} \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]

3. Calculate [tex]\( e^{0.16} \)[/tex]:

The value of [tex]\( e^{0.16} \)[/tex] is a constant that can be calculated using a calculator. For our purpose, it approximately equals 1.17351.

4. Compute the value of [tex]\( P \)[/tex]:

Now substitute the approximate value of [tex]\( e^{0.16} \)[/tex] into the equation:
[tex]\[
P \approx \frac{246.4}{1.17351} \approx 209.97
\][/tex]

5. Determine the closest value:

Comparing our calculated [tex]\( P \approx 209.97 \)[/tex] to the provided options, the closest approximate value is:

A. 210

So, the approximate value of [tex]\( P \)[/tex] is 210.

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